We derive optimality conditions for the optimum sample allocation problem, formulated as the determination of the fixed strata sample sizes that minimize the total cost of the survey, under assumed level of the variance of the stratified estimator and one-sided upper bounds imposed on sample sizes in strata. In this context, we take that the variance function is of some generic form that involves the stratified $\pi$ estimator of the population total with stratified simple random sampling without replacement design as a special case. The optimality conditions mentioned above will be derived with the use of convex optimization theory and the Karush-Kuhn-Tucker conditions. Based on the established optimality conditions we give a formal proof of the existing procedure, termed here as LRNA, that solves the allocation problem considered. We formulate the LRNA in such a way that it also provides the solution to classical optimum allocation problem (i.e. minimization of the estimator's variance under fixed total cost) under one-sided lower bounds imposed on sample sizes in strata. From this standpoint, the LRNA can be considered as a counterparty to the popular recursive Neyman allocation procedure that is used to solve the classical problem of optimum sample allocation but with one-sided upper bounds. Ready-to-use R-implementation of the LRNA is available through our package stratallo, which is published on the Comprehensive R Archive Network (CRAN) package repository.

翻译：本文推导了最优样本分配问题的最优性条件，该问题被表述为：在假设分层估计量方差水平及分层中样本容量单向上限约束下，确定使调查总成本最小化的固定分层样本容量。在此背景下，我们假定方差函数具有某种通用形式，该形式以分层简单随机无放回抽样设计下的总体总量分层π估计量作为特例。上述最优性条件将利用凸优化理论和Karush-Kuhn-Tucker条件推导得出。基于已建立的最优性条件，我们对当前称之为LRNA的现有流程给出了形式化证明，该流程可解决所考虑的分配问题。我们对LRNA进行了重新表述，使其在分层样本容量单向下限约束下，也能解决经典最优分配问题（即在固定总成本下使估计量方差最小化）。从这个角度看，LRNA可被视为流行的递归Neyman分配流程的对应方法，后者用于解决单向上限约束下的经典最优样本分配问题。LRNA的即用型R语言实现可通过我们发布的stratallo包获取，该包已在综合R档案网络（CRAN）包仓库中发布。